We present an analysis of high-resolution ($R ∼ 50,000$) infrared K-band echelle spectra of 14 T Tauri stars in the Orion Nebula Cluster. We model Zeeman broadening in three magnetically sensitive \ion{Ti}{1} lines near $2.2\ μ$m and consistently detect kilogauss-level magnetic fields in the stellar photospheres. The data are consistent in each case with the entire stellar surface being covered with magnetic fields, suggesting that magnetic pressure likely dominates over gas pressure in the photospheres of these stars. These very strong magnetic fields might themselves be responsible for the underproduction of X-ray emission of T Tauri stars relative to what is expected based on main-sequence star calibrations. We combine these results with previous measurements of 14 stars in Taurus and 5 stars in the TW Hydrae association to study the potential variation of magnetic-field properties during the first 10 million years of stellar evolution, finding a steady decline in total magnetic flux with age. ; Comment: 34 pages, 17 figures, published in ApJ, 2011, 729, 83